He is given many names in the series, including Morpheus and Oneiros (within Wikipedia he is generally referred to as Morpheus). It is known that the Endless have many aspects, one of which is the personification active at any one time, and if one aspect dies, another replaces it. One particular aspect of Dream is the central character of the series, so referring to this aspect of Dream as Morpheus differentiates him from Dream as a whole. When the aspect known as Morpheus dies at the end of The Kindly Ones, the ninth collection of issues in the series, he is replaced by a new aspect, which used to be a child of Morpheus' aspect called Daniel. This is a tricky concept, encapsulated in the tenth and final collection, The Wake, when one character at Morpheus' wake, perplexed by the question of who exactly has died, is told by another that the purpose of the wake is to mourn "a point of view". The other Endless remain personified by the same aspect throughout the series, so they are simply referred to by the generic names.
When interacting with individual mortals, he appears in a guise appropriate to the mortal. For instance, in the story "Tales in the Sand" he interacts with the ancestors of a black aboriginal tribe, and is depicted as a black man called Kai'Ckul. He is once also depicted as a cat, in the issue "Dream of a Thousand Cats", and once as partway between a cat and a human, when talking to the feline goddess Bast. In the story "Men of Good Fortune", Dream is seen at different times in the last 500 years; his costume is a little more conventional than the modern Dream, but still with an air of eccentricity. In one popular sequence in the issue "The Parliament of Rooks", he and his elder sister Death are depicted as cartoon-style children.
He invariably wears black, except when wearing his formal costume, which involves purple and blue. He has a helmet, which he seems to wear on occasions of great importance; this is his sigil in the galleries of the other Endless. Morpheus lives in a castle within his realm. Both the castle and the rest of the realm are mutable and change often, at Morpheus' will; but parts of both the castle and the realm are maintained in constant form as a courtesy to its inhabitants. It is perhaps significant that Morpheus is the only one of the Endless known to populate his realm - many other characters live there, including Cain and Abel. He even creates (and in some cases recruits) servants to perform roles he could easily carry out himself, including the reorganisation of the castle and the guarding of its entrance. This perhaps points at an essential loneliness in Morpheus' character.
Dream is a noble, tragic hero, very much in the traditional style of heroes of Greek tragedy. He is sometimes slow, a little at sea when dealing with humour, occasionally insensitive and often self-obsessed. (As Mervyn Pumpkinhead remarks, when one of Morpheus' invariably disastrous romances ends, "He's gotta be the tragic figure standing out in the rain, mournin' the loss of his beloved. So down comes the rain, right on cue. In the meantime everybody gets dreams fulla existential angst and wakes up feeling like hell. And we all get wet.") On the other hand, he is consistently aware of his responsibilties, both those to other people and those that go with his (for want of a better word) territory, which makes him both dependable and fair-minded. He shares a close, reciprocal bond of dependence and trust with his elder sister, Death. He consistently strives for understanding, most particularly of himself and of the other Endless, but is ultimately defeated by his most tragic flaw, his inability to consciously change himself and to recognise and accept the change that inevitably occurs at an unconscious level. As Lucien remarks in The Wake when asked (by Matthew, the raven) "Why did it happen? Why did he let it happen?", "Charitably...I think...sometimes, perhaps, one must change or die. And in the end, there were, perhaps, limits to how much he could let himself change."